Integral representations of solutions of the wave equation based on relativistic wavelets

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J. Phys. A: Math. Theor. 45 (2012) 385203 (14pp) doi:10.1088/1751-8113/45/38/385203

Integral representations of solutions of the waveequation based on relativistic wavelets

Maria Perel1,2 and Evgeny Gorodnitskiy1

1 Department of Mathematical Physics, Physics Faculty, St.Petersburg University, Ulyanovskaya1-1, Petrodvorets, St.Petersburg, 198504, Russia2 Ioffe Physical-Technical Institute of the Russian Academy of Sciences, 26 Polytekhnicheskaya,St. Petersburg 194021, Russia

E-mail: perel@mph.phys.spbu.ru

Received 18 November 2011, in final form 21 June 2012Published 31 August 2012Online at stacks.iop.org/JPhysA/45/385203

AbstractA representation of solutions of the wave equation with two spatial coordinatesin terms of localized elementary ones is presented. Elementary solutions areconstructed from four solutions with the help of transformations of the affinePoincare group, i.e. with the help of translations, dilations in space and time andLorentz transformations. The representation can be interpreted in terms of theinitial-boundary value problem for the wave equation in a half-plane. It givesthe solution as an integral representation of two types of solutions: propagatinglocalized solutions running away from the boundary under different anglesand packet-like surface waves running along the boundary and exponentiallydecreasing away from the boundary. Properties of elementary solutions arediscussed. A numerical investigation of coefficients of the decomposition iscarried out. An example of the decomposition of the field created by sourcesmoving along a line with different speeds is considered, and the dependence ofcoefficients on speeds of sources is discussed.

PACS numbers: 42.25.Bs, 02.30.Px, 11.30.Cp, 04.20.Jb, 02.60.Lj

1. Introduction

The construction of exact representations of solutions of the wave equation in termsof elementary localized solutions is the subject of this paper. Fourier analysis yields arepresentation of solutions in terms of plane waves. An analogue of Fourier analysis inour consideration is continuous wavelet analysis. We use here a special version of continuouswavelet analysis based on the transformations of the affine Poincare group [1] (wavelets basedon the Poincare group was discussed also in [2, 3]). The representation obtained by us isassociated with the initial-boundary value problem in the half-plane. It is given in terms of

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

the affine Poincare wavelet transform of time-dependent boundary data on a line, which is theboundary of the half-plane. The wavelet transform is efficient in the processing of functionsof time and space coordinate, which are results of observations and which have a multiscalestructure [1], [4]. The result of propagation of data is given by us in terms appropriate for dataprocessing.

Representations of solutions of the wave equation in terms of localized ones may beregarded as an initial step for the investigation of high-frequency wave propagation in aslowly inhomogeneous medium or of the diffraction on complex-shaped bodies. Asymptotictechniques involving localized solutions have been developed since the 1970s by Babich andPankratova [5], Popov [6], Babich and Ulin [7], Steinberg, Heyman and Felsen [8–10]. Themain point of the methods is the decomposition of Green’s function of the wave equationin an inhomogeneous medium in terms of Gaussian beams. To obtain the decompositioncoefficients, the slowly inhomogeneous medium is approximated by a homogeneous one nearthe source. The coefficients are calculated by the comparison of Green’s function asymptoticsat the infinity and the asymptotics obtained from the decomposition. The solution of the initial-boundary value problem (IBVP) in a half-plane (or half-space) is reduced to Green’s functiondecomposition by the Kirchhoff formula. The decomposition is widely used in seismology[11–15]. The method is applicable near caustics unlike the ray method. Another advantage islocality; local changes of the medium impact only a few Gaussian beams in the decomposition,which pass through the changed area.

Considering a homogeneous medium is a required step for further development of amethod for the IBVP that involves localized solutions from a wide class.

The exact representations of solutions of the non-stationary wave equation in ahomogeneous medium as superpositions of elementary localized solutions have been presentedfor the first time in the work of Kaiser [16]. They were based on the analytic wavelet transform,and he used very special spherically symmetric elementary solutions. Representations ofsolutions in terms of elementary solutions from a wide class were proposed in [17–19]and discussed in [20]. These representations were based on continuous wavelet analysis.Elementary solutions were constructed from two chosen ones by means of a similitude groupof transformations that contains shifts, scalings and rotations. Representations based on thediscreet version of wavelet analysis were constructed by Quan and Ying [21].

In this work, a solution of the non-stationary boundary problem for the wave equationhas been decomposed in localized elementary solutions by means of the space–time versionof wavelet analysis. Elementary solutions are constructed from four chosen ones by meansof an affine Poincare group of transformations that contains shifts and scaling both in spaceand in time and the Lorentz transformations. Preliminary results were reported in [22]. Theaffine Poincare wavelet transform (APWT) is a coefficient in the decomposition of solutions.The region of integration is determined by the values of parameters under which the APWT isabove a certain level. A numerical implementation and examples of the calculation of APWTwere discussed in our work [23].

The method presented here has the following advantages. The formulas obtained areexact in a homogeneous medium and satisfy boundary conditions. In contrast, methods ofthe Gaussian beams summation do not reproduce boundary conditions exactly. Our methodcan utilize localized solutions in a wide class. The possibility of taking solutions from a wideclass enables one to build a flexible problem-adapted representation. In particular, this meansthat one can construct a sparse representation of the boundary data for a particular problem,which decreases the number of necessary elementary solutions. The wavelet techniques allowone in a natural way to extract the most interesting features of the wave field, for example,the wave fronts without calculating the whole field. This feature was demonstrated by us

2

J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

in [20]. This research allows for further semi-analytic investigations of waves in a slowlyinhomogeneous medium: the high-frequency part may be studied by asymptotic methods, thelow-frequency part of the field may be calculated numerically. In summary, we have developeda method, which is suitable not only for the development of new numerical schemes, but alsofor qualitative research.

We hope that our results may be useful also in astronomy to determine the velocities ofrelativistic objects.

For the sake of simplicity we consider here only the case of two spatial dimensions.In section 2, we recall APWT theory. In section 3, we develop Poincare wavelet theory

for solutions and construct representations of solutions of the wave equation based on it.Then we give examples of mother solutions and discuss properties of the elementary solutionsobtained from them. Elementary solutions are built by using the Lorentz transformations,which are associated with fields in different frames moving with relativistic speeds. However,in problems of acoustic wave propagation such an interpretation is not applicable. We showthat the parameter of the speed is closely connected with the outgoing angle of the packet. Animplementation and interpretation of Poincare wavelet transform are considered in section 5with several examples.

We suggest to determine mother wavelets by means of their Fourier transforms, since wedo not have their analytic form in the coordinate domain. However, we may take as mothersolutions exact particle-like solutions found in [24–27], which are given by simple explicitformulas both in the coordinate and in the Fourier domains. The review of other exact localizedsolutions is given in [28, 29]. Solutions obtained by Lorentz transformations are discussed in[30].

2. Basics of affine Poincare wavelet analysis

The first steps in the application of wavelet analysis methods are the definition of a class offunctions H considered, a choice of a mother wavelet ψ ∈ H and the construction of a familyof wavelets. The mother wavelet is a fixed function. A family of wavelets consists of functionsobtained from the mother wavelet by means of transformations of a group. The set of waveletsshould be dense in the chosen space H. Then it should be proved that any function f ∈ H canbe represented as an integral superposition of wavelets.

It is useful to consider together functions and their Fourier transforms. Below we introducethe notation

χ =(

ct

x

), σ =

(ω/c

kx

), (1)

and define the Minkovsky inner product as follows: (σ,χ) = −ωt + kxx. The method ofwavelet analysis itself is not associated with the wave equation. We apply it to the boundarydata function f (χ) ≡ f (ct, x) dependent only on one spatial coordinate x and time t.3 TheFourier transform of any f (χ) ∈ L2 reads

f (σ) =∫

R2d2χ f (χ) e−i(σ,χ). (2)

3 Here c is a parameter of the dimension of speed. We use the notation, taking in mind further applications to thewave equation in two spatial dimensions where the wave number is k = |k|, the frequency is ω = ±ck and c is thespeed of light.

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

Lorentz transformations enables one to connect the coordinate x and the time t in a stationarycoordinate system with the coordinate x′ and the time t ′ in a moving one with a speed v. Theyread (

ct ′

x′

)= �φ

(ct

x

), (3)

�φ =(

cosh φ − sinh φ

− sinh φ cosh φ

), tanh φ = v

c, (4)

where φ is called the rapidity and v is the speed of the moving frame. The frequency andthe wave number in the moving frame and in the stationary frame given by vectors σ ′ and σ,

respectively, are related as σ ′ = �−φσ.

We are going to clarify the choice of the spaceH. We choose any ψ(ct, x) ∈ L2. Constructa family of wavelets with transformations of the affine Poincare group, i.e. with shifting, scalingand the Lorentz transformations:

ψμ(χ) = 1

aψ

(�−φ

χ − ba

), b =

(cτ

bx

), (5)

μ = {b, a, φ}. (6)

A wavelet ψμ(χ) for a = 1, b = 0 represents a wave packet in the stationary frame, such thatit is the mother wavelet in the moving frame. The constructed family is not dense in the wholeL2. To show this, we analyze the Fourier transform of wavelets

ψμ(σ) = aψ (a�−φσ) e−i(σ,b) (7)

and show that we cannot go beyond the region bounded by the lines ω = ±ckx under the actionof Lorentz transformations. For example, a point from the domain k > 0, |kx| < k cannot bemapped to any point in the domain kx > 0, |k| < kx, by choosing a and φ.

The explanation of this fact is given below. Let the support of the Fourier transform of themother wavelet in the moving frame be concentrated near a point σ ′

0, i.e. ψ (σ ′) ≡ 0 outside aneighborhood of σ ′

0. The support of this packet in the stationary frame lies in the neighborhoodof σ0, such that σ0 = a−1�φσ ′

0. It is easy to check that the point (ω0/c, kx0) lies on thehyperbola (ω0/c)2 − k2

x0 = a−2((ω′0/c)2 − k

′2x0). For any prescribed (ω0, kx0) and (ω′

0, k′x0),

we can find a unique parameter a > 0. The parameter φ can be found by the relationtanh(φ + φ′) = ckx0/ω0, where tanh(φ′) = ck′

x0/ω′0. As φ increases, the point (ω0/c, kx0)

tends to the asymptotes of the hyperbola ω = ±ckx. Let ω′0 > 0, c|k′

x0| < ω′0. Then for any a

and φ, the point (ω0/c, kx0) lies in the same region: ω0 > 0, c|kx0| < ω0.

Therefore, we should decompose the space L2(R2) into a direct sum: L2(R

2) =4⊕

j=1H j

taking into account their Fourier transforms. The space H j comprises functions f j ∈ L2(R2),

which have the support of their Fourier transform in D j (see figure 1) :

f j(χ) = 1

(2π)2

∫D j

d2σ f (σ) ei(σ,χ). (8)

The mother wavelet is any function ψ j(χ) ∈ H j that satisfies an admissibility condition:

Cj =∫D j

d2σ|ψ j(σ)|2

|(ω/c)2 − k2x |

< ∞. (9)

The family of wavelets is constructed by formulas (5).

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

(a) Domains D1 and D2 (b) Domains D3 and D4

Figure 1. Domains D j that are supports of the Fourier transform of functions from H j ,j = 1, 2, 3, 4. The set of points obtained by Lorentz transformations from the point with φ = 0is a hyperbola. Arrows on hyperbolas show the direction of point movement from φ = −∞ toφ = +∞.

The APWT Fj(μ) of the function f j is defined as follows [1]:

Fj(μ) =∫

R2d2χ f j(χ)ψ jμ(χ), μ = {b, a, φ}. (10)

We denote functions from L2 by Latin lowercase letters and their wavelet transforms by Latinuppercase letters. It is not necessary to extract f j(χ) from f (χ) to calculate Fj(μ), as followsfrom the Plancherel equality:

( f j, ψ jμ) = 1

(2π)2( f j, ψ jμ) = 1

(2π)2( f , ψ jμ) = ( f , ψ jμ). (11)

The main fact that yields the decomposition of solutions is the Plancherel equality for theAPWT [1]:

( f j, g j)L2 = 1

Cj

∫dμ Fj(μ)Gj(μ), (12)

where ∫dμ ≡

∫R

dφ

∫ ∞

0

da

a3

∫R2

d2b. (13)

Equality (12) is valid for any f j, g j ∈ H j if the admissibility condition (9) is fulfilled.The reconstruction formula following from the Plancherel equality reads

f j(χ) = 1

Cj

∫dμ Fj(μ)ψ jμ(χ). (14)

See the appendix for more detail. Any f (χ) ∈ L2 can be represented as f (χ) =4∑

j=1f j(χ).

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

3. Decomposition of solutions

In the present section, we give exact decompositions of solutions of the wave equation basedon the formulas discussed above. The decomposition is associated with the boundary problemfor the wave equation in a half-plane y � 0, −∞ < x < ∞:

utt = c2(uxx + uyy), (15)

u(ct, x, y) |y=0 = f (ct, x), (16)

u(ct, x, y) |t�0 = 0. (17)

where c is the speed, t is the time and x and y are spatial variables. The function f (ct, x) in(16) is assumed to be square integrable:∫

R

d(ct)∫

R

dx | f (ct, x)|2 < ∞. (18)

The desired decomposition is expressed in terms of solutions running away from the boundaryunder different angles. It was first reported in [22].

We define four subspaces of solutions H j, j = 1, . . . , 4 of the wave equation using theFourier transform of the boundary data f (σ). Any function u j(χ, y) ∈ L2 belongs to H j if itsFourier decomposition reads

u j(χ, y) =∫D j

d2σ f (σ) ei(σ,χ)ei√

k2−k2x y, j = 1, 2, (19)

uj(χ, y) =∫D j

d2σ f (σ) ei(σ,χ)e−√

k2x −k2y, j = 3, 4, (20)

where k = ω/c is the wave number for any f (σ) ∈ L2. The functions u j(χ, y), j = 1, . . . , 4,

are solutions of the wave equation. If a function f (σ) decreases at infinity too slowly toyield a classical solution, the functions uj(χ, y) satisfy the wave equation in the distributionsense. The solutions u j(χ, y), j = 1, 2, are propagating, the solutions uj(χ, y), j = 3, 4 areexponentially decreasing and associated with details of the spatial behavior of f , which aresmaller than the wavelength. Note that the reconstruction formula for the Fourier transformyields

∑4j=1 u j(χ, 0) = f (χ).

In what follows we will use the Fourier transforms of solutions with respect to t and xdenoted by u j(σ; y) :

u j(σ; y) = f (σ) ei√

k2−k2x y, j = 1, 2, (21)

u j(σ; y) = f (σ) e−√

k2x −k2y, j = 3, 4. (22)

The Fourier transforms of solutions with respect to x and y dependent on t are denoted byu j(ct; kx, ky):

u j(ct; kx, ky) = f (σ) e−iωt, j = 1, 2, σ = (√k2

x + k2y , kx

)�. (23)

The solutions u j(ct, x, y), j = 3, 4, increase exponentially for negative y. The Fouriertransform of them with respect to x and y do not exist.

To develop a procedure of continuous wavelet analysis in the subspace H j, we shouldchoose a solution j ∈ H j. We call it the mother solution. The notation of the mother solution

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

begins with the Greek uppercase letter and that distinguishes it from the mother wavelet inL2, which is denoted by the Greek lowercase letter. The mother solution can be reduced tothe mother wavelet if y = 0. Therefore, ψ j(χ) = j(χ, 0) should satisfy the admissibilitycondition (9).

Then we construct a family of solutions:

jμ(χ, y) = 1

a j

(�−φ

χ − ba

,y

a

). (24)

Each solution represents a wave packet considered in the stationary frame, which is a shiftedand scaled mother solution in the frame moving in the x direction. These solutions aretransformed to wavelets from family (5) on the line y = 0, i.e. jμ(χ, 0) ≡ ψ jμ(χ), whereψ jμ(χ) are wavelets.

In terms of the mother wavelets, the solutions of (15), (17) may be decomposed as follows:

u(χ, y) =4∑

j=1

u j(χ, y), u j(χ, y) = 1

Cj

∫dμ Fj(μ) jμ(χ, y), (25)

Fj(μ) =∫

R2d2χ f (χ) jμ(χ, 0). (26)

It is a solution because it is a superposition of solutions. It is reduced to the reconstructionformula for the boundary data f if y = 0 and therefore it satisfies the boundary conditions.Each of these solutions moves or decreases away from the boundary. We do not take intoaccount solutions that come from infinity and increase in the positive y direction to satisfyinitial conditions (17).

The contribution of each solution is determined by the coefficient Fj, which is the wavelettransform of the boundary data.

4. Examples of mother solutions

Mother solutions can be taken from a wide class. We give examples of mother solutions,which behave as particles. As a starting point we take a solution that has the Fourier transformlocalized near the point kx = 0, ky = :

(ct; kx, ky) = exp

(−σ 2

‖ (ky − )2

2− σ 2

⊥k2x

2

)e−iωt, (27)

where ω = c|k|. If t = 0, then the solution in the coordinate domain reads

(0, x, y) = 1

2πσ‖σ⊥exp

(iy − y2

2σ 2‖

− x2

2σ 2⊥

). (28)

If t = 0, the solution cannot be found explicitly. But if t is small enough, then 1(ct, x, y) ≈1(0, x, y − ct). It can be obtained by expanding k =

√k2

x + k2y up to linear terms:

ω = ω0 + c(ky − ) + . . . and substituting it into the phase:

kxx + kyy − ωt = kxx + ky(y − ct) + · · · . (29)

In this way, for small time we obtain a packet-like solution which is filled with oscillationsof a spatial frequency and which has the Gaussian envelope. It moves along the y axis withspeed c.

For large t, the solution should be found numerically. Exact solutions having similar localbehavior were found in [25], [26] and were named ‘Gaussian wave packets’. Solutions and

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

0 5 10 15 20

0

5

10

15

x

y

(a) Solutions in the moving frame

0 5 10 15 20

0

5

10

15

x

y

(b) Same solutions in the stationary frame, φ = 1

Figure 2. Real part of mother wavelet 1 with parameters = 16, σ‖ = √2, σ⊥ = √

2 insuccessive time moments: ct = 0, ct = 7.5, ct = 15.

their Fourier transforms were given by explicit formulas and were studied analytically. Aninvestigation of these solutions from the point of view of continuous wavelet analysis wasperformed in [27].

To be a mother solution in H1, this solution should contain Fourier components withω > 0 and ky > 0. The last condition is fulfilled only approximately if the central frequencyof the packet is larger than the width of the support of the Fourier transform in the ky

direction 1/σ‖, i.e. σ‖ � 1. To obtain the mother solution, we assume that formula (27) isvalid only for ky > 0 and (ct; kx, ky) = 0 if ky � 0. To make the Fourier transform smooth,we introduce an additional factor exp (− 1

ky), ky = √

k2 − k2x . The mother solution 1(ct, x, y)

is of the form

1(ct, x, y) = 1

(2π)2

∫ ∞

−∞dkx

∫ ∞

0dky exp

(−σ 2

‖ (ky − )2

2− σ 2

⊥k2x

2

)× exp(−1/ky) exp

(i(kxx + kyy − kct)

), (30)

k = |k|. Because of the factor exp (−1/ky), the solution satisfies the admissibility condition.

We take a variable of integration k =√

k2x + k2

y instead of ky and write the Fourier transformof (30) in the form of (21). Then we obtain

1(σ; y) = k

kyexp

(−σ 2

‖ (ky − )2

2− σ 2

⊥k2x

2− 1

ky

)· eikyy, (31)

ky = √k2 − k2

x . Figure 2(a) shows this solution in successive time moments. It representsa wave packet moving away from the boundary y = 0. It has a finite energy because it islocalized in an exponential way. The same solution for other choice of parameters is shown infigure 3(a).

The mother solution in H3 is constructed in a similar way. We interchange k and kx, putiy instead of y, and obtain

3(σ; y) = kx√k2

x − k2exp

(−σ 2

‖ (√

k2x − k2 − )2

2− σ 2

⊥k2

2

)× exp

( − 1/

√k2

x − k2) · e−

√k2

x −k2y. (32)

The solution 3(ct, x, y) is a wave packet running along the boundary and decreasingexponentially in the positive y direction. The Fourier transform of the solution comprises

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

0 5 10 15 20

0

5

10

15

x

y

0 5 10 15 20

0

5

10

15

x

y

(a) Solutions in the moving frame (b) The same solutions in the stationaryframe, φ = 1

Figure 3. Real part of the mother wavelet 1 with parameters = 4, σ‖ = 1, σ⊥ = 2 in successivetime moments: ct = 0, ct = 7.5, ct = 15.

(a) Solutions in the moving frame (b) The same solutions in the stationary frame,φ = 1

Figure 4. Real part of the mother wavelet 3 with parameters = 4, σ‖ = 1, σ⊥ = 2 in successivetime moments: ct = 0 and ct = 7.5.

components with wavelengths in the x direction, which are smaller than the wavelength invacuum. Figure 4(a) demonstrates the solution in successive time moments. The waveletanalysis of solutions from H3 and H4 might be useful for analyzing surface waves, but a moredetailed study of the problem goes beyond the scope of this paper.

The mother solutions in H2 and H4 can be obtained as follows: 2(ct, x, y) =1(−ct, x, y), 4(ct, x, y) = 3(ct,−x, y).

Now we discuss analytically the properties of the family of solutions obtained bymeans of Lorentz transformations. The mother solution 1(ct, x, y) represents a wavepacket moving in the positive y direction and localized near the point x = 0, y = ct. Inthe Fourier domain, it is localized near the point �σ0 = (, 0), ω0 = c. Now consider1(ct ′, x′, y), where x′, y and t ′ are coordinates in the frame that moves with speed v andrapidity φ. The coordinates in the moving frame are connected with the coordinates in thestationary frame by transformations (3), (4). Assuming that t ′ is small enough, we obtain1(ct ′, x′, y) ≈ 1(0, x′, y − ct ′) = 1(0, cosh φ(x − ct tanh φ), y + sinh φx − cosh φct).

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

Figure 5. The support of the wavelet in the Fourier domain for various φ

The position of the center of the packet x0 and y0 in the stationary frame is x0 = ct tanh φ,

y0 = ct/ cosh φ. Packet (30) found in the moving frame is numerically small outside an ellipse

(y − ct ′)2

2σ 2‖

+ x′2

2σ 2⊥

� 1. (33)

For the sake of simplicity, we assume that it is a circle σ ≡ σ‖ = σ⊥:

(x′)2 + (y − ct ′)2 � 2σ 2. (34)

An example of such a solution is shown in figure 2(a). Taking into account (3) we find thedomain in the stationary frame corresponding to the circle (34) in the moving frame. It is anellipse:

cosh2 φ(x − x0)2 + ((y − y0) + sinh φ(x − x0))

2 � 2σ 2. (35)

This ellipse in the main axes reads

λ1x2 + λ2y2 � 2σ 2, (36)

where

λ1,2 = e±φ cosh φ. (37)

The directions of the main axes are

e1,2 =(±e∓φ

1

). (38)

The new coordinates x and y are connected with the old coordinates x − x0 and y − y0 by therelation (

x

y

)= U�

(x − x0

y − y0

), U =

(e1

|e1| ,e2

|e2|)

, (39)

where the columns of the matrix U are vectors e j, j = 1, 2.

The axes of the ellipse in the stationary frame are not orthogonal to the direction ofpropagation and to wave fronts (see figure 2).

Similar considerations are valid in the Fourier domain. The support of the Fouriertransform of the solution is distorted under the action of the Lorentz transform. Examplesof calculations for various φ are given in figure 5; for a discussion of figure 5 see below.

5. Numerical calculations of coefficients in the wavelet decompositions

According to (25), each solution is decomposed into a sum of fourfold integrals over μ.

Here, we show that calculations of such an integral can be efficient for some boundary data.For realistic signals, the domain of integration may be small. The domain of integration is

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J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

(a) The function f(ct, x) (b) The scale-rapidity diagram S1(a, φ)

Figure 6. An example of the scale-rapidity diagram.

determined by the numerical support of the coefficient in decomposition (25), which is thewavelet transform of boundary data. The wavelet transform is non-negligible if the supportsof a signal (the domain where the signal is non-zero) and of a wavelet from family (5) have anintersection. The supports of Fourier transforms of a signal and a wavelet should also have anintersection.

The support in the Fourier domain is governed by a scale and a rapidity. Figure 5demonstrates transformations of the support of the wavelet in the Fourier domain for differentvalues of the rapidity φ. The support for φ = 0 is a circle. Upon the Lorentz transformations,the support takes the form of an ellipse and its center moves along the hyperbola. Here, there isan analogy with the transformations of the support in a spatial domain. A change in the scalingparameter a will lead to a change in the hyperbola, where the centers of supports are located.

Shifts in the spatial domain b rule oscillations of an integrand in (10). An increase inoscillations results in a decrease in the wavelet transform and vice versa. If the support of awavelet for some parameters matches the support of a signal and b is taken in such a way thatoscillations are minimized, then the wavelet transform has a maximum.

Now consider an example. We assume that a function f represents a field that is generatedin the plane y = 0 by six groups of monochromatic point sources, which move in the x directionwith different speeds in plane y = −5000. In each group, the speeds are distributed with respectto the Gaussian law with σ = 0.01c, σ is a dispersion of the distribution and c is the speed inthe wave equation, c = 1. The sources have different frequencies. The sources in the jth group,j = 1, . . . , 6, are characterized by the frequency ω j and the rapidity φ j, which corresponds tothe mean speed. We choose the parameters as follows: ω1 = 1, φ1 = 0.4; ω2 = 1, φ2 = 0.7;ω3 = 1, φ3 = 0.5; ω4 = 0.9, φ4 = 0.3; ω5 = 0.95, φ5 = 0.5; ω6 = 0.95, φ6 = 0.4. Theboundary function f as a function of x and t is shown in figure 6(a).

We show that, analyzing the field in the plane y = 0 by means of the wavelet transform, wecan determine the parameters of the sources. The sources are far from the observation plane,and we do not calculate the wavelet transforms in domains D3,4 because the contribution ofthe components propagating along the surface are negligible.

We show that the wavelet transform enables us to find the frequencies and speeds ofsources. We are not interested in the determination of the position of sources; for this reason,we introduce a function S(a, φ), which reads

S1(a, φ) = 1

a3

∫R2

d2b |F1(μ)|2, μ = {b, a, φ}. (40)

11

J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

Such a function, corresponding to the field in figure 6(a), as a function of the scaling a andthe rapidity φ is given in figure 6(b). The scale-rapidity diagram shown is calculated withthe following wavelet parameters: = 4, σ‖ = 2

√55 ≈ 15 and σ⊥ = 8, the calculation

mesh is from −128 to 128 with step 0.5 in both x and ct. We see six maxima in this figure,which correspond to each group of sources. The position of the maximum a and φ containsinformation about the frequency ω = 1/a and the average rapidity φ of sources. (We recallthat the speed v is connected with φ by the relation v/c = tanh φ.) The domain of integrationin the decomposition formula must contain values of the parameters a and φ such that S isnot small. If we are interested only in the investigation of sources of certain frequencies andrapidities, we need to take the corresponding domain of integration.

A more complicated situation where the sources are masked by noise are considered byus in [23]. It is shown therein that the contribution of a noise can be eliminated because it islocated at small scales on the scale-rapidity diagram. The parameters of sources are found.

Acknowledgments

EG is indebted to Dr Ru-Shan Wu for his hospitality at the University of California, Santa-Cruzwhich made possible the completion of the text and figures preparation.

Appendix. The reconstruction formula

We prove the Plancherel equality (12) for the affine Poincare wavelet transform (10) to makethe paper self-contained. For the sake of brevity, we assume that f ≡ g. With account of (13),the Plancherel equality reads∫

R

dφ

∫ ∞

0

da

a3

∫R2

d2b |Fj(b, a, φ)|2 = Cj

∫R2

d2χ| f j(χ)|2, (A.1)

where Cj is defined by relation (9).Applying the Plancherel equality (11) to the inner integral with respect to b and taking

into account the definition of Fj, we obtain∫R2

d2b |Fj(b, a, φ)|2 = a2

(2π)2

∫R2

d2σ | f (σ )|2 |ψ j(a�−φσ)|2. (A.2)

Here we use the fact that Fj is a convolution of two functions f (χ) and ψ jμ(−χ). Then we

assume that the Fourier transform of ψ jμ(−χ) is ψ jμ(σ), which is given by formula (7).Substituting (A.2) on the left-hand side of (A.1) and interchanging the order of integration,

we obtain∫R

dφ

∫ ∞

0

da

a3

∫R2

d2b |Fj(b, a, φ)|2 = 1

(2π)2

∫R2

d2σ| f (σ)|2∫

R

dφ

∫ ∞

0

da

a|ψ j(a�−φσ)|2

(A.3)

The next step is an analysis of the inner integral on the right-hand side of (A.3). For the sake ofdefiniteness, we assume that j = 1. We transform the variable of integration in the followingway:

σ ′ = a�−φσ = a

(k cosh φ + kx sinh φ

k sinh φ + kx cosh φ

)= a

(ρ cosh(φ + φ0)

ρ sinh(φ + φ0)

)= ρ ′

(cosh φ′

sinh φ′

). (A.4)

Here, we use the notation: k = ρ cosh φ0, kx = ρ sinh φ0, ρ > 0. Instead of a and φ we takenew variables φ′ = φ + φ0 and ρ ′ = aρ, φ′ ∈ (−∞,+∞), ρ ′ ∈ (0,∞). Then the vector

12

J. Phys. A: Math. Theor. 45 (2012) 385203 M Perel and E Gorodnitskiy

variable σ ′ is chosen instead of φ′ and ρ ′. It takes values in D1. The inner integral for anyfixed σ is transformed as follows:∫

R

dφ

∫ ∞

0

da

a|ψ j(a�−φσ)|2 =

∫R

dφ′∫ ∞

0

ρ ′dρ ′

ρ ′2 |ψ j(σ′)|2 (A.5)

=∫

R2d2σ ′

∣∣ψ j(σ′)∣∣2

k′2 − k′2x

≡ Cj, σ ′ =(

k′

k′x

). (A.6)

Formula (12) has a generalization, in which f and g are any functions from L2 and the wavelettransforms of the functions f and g denoted by Fj and Gj, respectively, are calculated withdifferent mother wavelets, say ζ and ψ, respectively. In this case, formula (12) is valid, but Cj

is calculated as follows:

Cj =∫D j

d2σζ j(σ)ψ j(σ)

|(ω/c)2 − k2x |

< ∞. (A.7)

The reconstruction formula (14) may be obtained from (12) formally if we choose as a functiong any sequence of functions that tends to the Dirac δ function. To pass to the limit under thesign of the integral in (12), we must impose additional restrictions on the functions f and g.

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